Analysis and Parametric Synthesis of the Piano Sound

HELSINKI UNIVERSITY OF TECHNOLOGY

Department of Electrical and Communications Engineering

Laboratory of Acoustics and Audio Signal Processing

Heidi-Maria Lehtonen

Analysis and Parametric Synthesis of the Piano Sound

Master’s Thesis submitted in partial fulfillment of the requirements for the degree of

Master of Science in Technology.

Espoo, November 29, 2005

Supervisor: Professor Vesa Välimäki

Instructor: M.Sc. Jukka Rauhala

HELSINKI UNIVERSITY ABSTRACT OF THE

OF TECHNOLOGY MASTER’S THESIS

Author: Heidi-Maria Lehtonen

Name of the thesis: Analysis and Parametric Synthesis of the Piano Sound

Date: Nov 29, 2005 Number of pages: 55

Department: Electrical and Communications Engineering

Professorship: S-89 Acoustics and Audio Signal Processing

Supervisor: Prof. Vesa Välimäki

Instructors: M.Sc. Jukka Rauhala

In this thesis, an overview of the sound production mechanism of the piano is given. The

acoustical properties of the instrument are studied in order to make a baseline for a physical

and parametric model for the piano. In addition, the most important features of the piano

sound, such as inharmonicity, the complicated decay process of the tones and the properties of

the soundboard and the pedals, are investigated. The differences between the grand piano and

the upright piano are considered in brief.

As the digital waveguide technique is the most feasible physics-based sound synthesis tech-

nique at the moment, the synthesis procedure that is followed in this thesis is based on this

technique. An overview of the main aspects of this synthesis scheme is given, and the most

important modeling issues are taken into account from the piano sound synthesis point of view.

A novel filter design technique for modeling the losses occurring in the piano sound is pre-

sented with some practical design examples. In addition, the modeling of the sustain pedal

is discussed and signal analysis is performed in order to gather information for the synthetic

sustain pedal algorithm. The analyzed signals are obtained from two recording sessions which

were carried out in two parts during the year 2005.

Keywords: digital filter design, digital signal processing, music, musical acoustics, piano, sig-

nal analysis, sound synthesis

i

TEKNILLINEN KORKEAKOULU DIPLOMITYÖN TIIVISTELMÄ

Tekijä: Heidi-Maria Lehtonen

Työn nimi: Pianon äänen analyysi ja parametrinen synteesi

Päivämäärä: 29.11.2005 Sivuja: 55

Osasto: Sähkö- ja tietoliikennetekniikka

Professuuri: S-89 Akustiikka ja äänenkäsittelytekniikka

Työn valvoja: Prof. Vesa Välimäki

Työn ohjaaja: DI Jukka Rauhala

Tässä työssä tutkitaan pianon äänentuottomekanismia sekä akustisia ominaisuuksia. Tarkoi-

tuksena on luoda lähtökohdat pianon äänen parametriselle mallintamiselle. Lisäksi tutkitaan

pianon äänen tärkeimpiä ominaisuuksia, kuten epäharmonisuutta, osaäänesten monimutkaista

vaimenemisprosessia, kaikupohjan ja pedaalin ominaisuuksia sekä näiden tekijöiden vaikutuk-

sia ääneen. Flyygelin ja pystypianon eroja tarkastellaan lyhyesti.

Koska digitaalinen aaltojohtomallinnus tarjoaa parhaat lähtökohdat fysikaaliseen soitinmallin-

nukseen, tämä työ pohjautuu tähän tekniikkaan. Digitaalisen aaltojohtomallinnuksen pääpiir-

teet esitellään, kuten myös pianon kannalta olennaisimmat mallinnukseen liittyvät asiat. Lisäksi

esitellään uusi tekniikka häviösuotimen suunnittelua varten, sekä annetaan muutama esimerk-

ki käytännön suodinsuunnittelusta tällä tekniikalla. Tämän lisäksi tarkastellaan kaikupedaalin

mallintamista sekä suoritetaan signaalianalyysi tehokkaan mallinnusalgoritmin löytämiseksi.

Analysoitavat signaalit on äänitetty kahdessa äänityssessiossa vuoden 2005 aikana.

Avainsanat: digitaalinen signaalinkäsittely, digitaalisuotimien suunnittelu, musiikki, musiik-

kiakustiikka, piano, signaalianalyysi, äänisynteesi

ii

Acknowledgements

This Master’s thesis has been carried out in the Laboratory of Acoustics and Audio Signal

Processing at Helsinki University of Technology during the year 2005. The work has been

funded by the Academy of Finland as a part of the CAPSAS project (project no. 104934).

My foremost thanks go to my supervisor, Professor Vesa Välimäki, whose encouragement

and many helpful comments have contributed significantly to my whole working process.

Also, I want to thank my instructor M.Sc. Jukka Rauhala, who has given me many valuable

advises during this work.

I wish to thank my colleagues at the Akulab, who have created a pleasant working environ-

ment. Especially, I would like to thank M.Sc. Henri Penttinen for providing me informa-

tion about recording techniques, Mr. Antti Huovilainen for helping me with coding-related

problems and Mr. Jussi Pekonen for helping me to carry out the recordings. I would also

like to thank Mrs. Lea Söderman for taking care of all practical issues.

Finally, I would like to thank my family for their love and prayers. My father, Professor

Pentti Huovinen, has encouraged me to believe in myself and in my dreams. My mother, Dr.

Saara Huovinen, has supported me in every aspects of my life during this whole process.

My brother Ville and sister Elisa, thank you for your encouragement. I would also like to

thank my parents-in-law for their support.

Last, but definitely not least, I would like to thank my dear husband Hannu. Thank you for

your patience and love during the good and especially the hard times.

Otaniemi, November 29, 2005

Heidi-Maria Lehtonen

iii

Contents

Abbreviations vi

List of Figures vii

1 Introduction 1

2 Overview of the Piano 3

2.1 Keyboard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Action and the Hammer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Piano Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Inharmonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.2 Sound Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 The Soundboard and the Frame . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4.1 Pedals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.5 Grand Piano vs. Upright Piano . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5.1 Structural Properties of the Upright Piano . . . . . . . . . . . . . . 10

2.5.2 Is There Noticeable Difference Between the Sounds of the Two In-

struments? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Digital Waveguide Modeling of the Piano 13

3.1 A Short Overview of Synthesis Methods . . . . . . . . . . . . . . . . . . . 13

3.1.1 Abstract Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 13

iv

3.1.2 Processed Recordings . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.3 Spectral Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.4 Physics-based Modeling Techniques . . . . . . . . . . . . . . . . . 14

3.2 Digital Waveguide Modeling of the Piano . . . . . . . . . . . . . . . . . . 15

3.2.1 Modeling the String . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.2 Modeling the Hammer . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.3 Modeling the Soundboard . . . . . . . . . . . . . . . . . . . . . . 21

4 A Novel Loss Filter Design Technique 22

4.1 Equalizer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Anti-Imaging Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3 Multi-Ripple Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.4 Design Example: Perfect Match of 50 Partials . . . . . . . . . . . . . . . . 26

4.5 Design Example: Low Order Sparse FIR Filter . . . . . . . . . . . . . . . 26

4.6 Results and Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Modeling of the Sustain Pedal 30

5.1 Overview of Prior Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.1.1 The Sustain Pedal Effect . . . . . . . . . . . . . . . . . . . . . . . 31

5.1.2 Studies Concerning the Modeling of the Sustain Pedal . . . . . . . 31

5.2 Reverberator Algorithm for Sustain Pedal Modeling . . . . . . . . . . . . . 32

5.2.1 Signal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2.2 Algorithm for Modeling the Sustain Pedal Effect . . . . . . . . . . 35

5.2.3 Results and Analysis of the Algorithm . . . . . . . . . . . . . . . . 41

5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6 Conclusions and Future Directions 42

A Recordings 50

v

Abbreviations

DFT Discrete Fourier Transform

FDN Feedback Delay Network

FFT Fast Fourier Transform

FIR Finite Impulse Response

FM Frequency Modulation

IFFT Inverse Fast Fourier Transform

IFIR Interpolated Finite Impulse Response

IIR Infinite Impulse Response

KS Karplus-Strong

LPC Linear Predictive Coding

VCO Voltage Controlled Oscillator

vi

List of Figures

2.1 A grand piano. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 An upright piano. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 The general structure of a grand piano. . . . . . . . . . . . . . . . . . . . . 5

3.1 Two parallel delay lines for simulating two propagating waves. . . . . . . . 16

4.1 The block diagram of the�

th-order loss filter. . . . . . . . . . . . . . . . . 23

4.2 The magnitude responses of the subfilters presented in Fig. 4.1. . . . . . . . 24

4.3 The magnitude response and the reverberation times of the loss filter. . . . . 27

4.4 Comparison of different loss filters. . . . . . . . . . . . . . . . . . . . . . 28

5.1 3D plot of the string register response, the key index is 4. . . . . . . . . . . 33

5.2 3D plot of the string register response, the key index is 40. . . . . . . . . . 33

5.3 3D plot of the string register response, the key index is 76. . . . . . . . . . 34

5.4 The energies of the string register responses with and without the sustain

pedal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.5 The envelopes of the harmonics 2-7 of the tone C1. . . . . . . . . . . . . . 37



5.8 The block diagram of the sustain pedal algorithm. . . . . . . . . . . . . . . 39

A.1 Microphone positioning, grand piano. . . . . . . . . . . . . . . . . . . . . 51

A.2 Microphone positioning, upright piano. . . . . . . . . . . . . . . . . . . . 52

A.3 Recording the soundboard response. . . . . . . . . . . . . . . . . . . . . . 52

vii

Chapter 1

Introduction

In this thesis, the aim is to analyze the piano sound and study the modeling of the instrument.

The piano is one of the most widely used instrument in the western music. Moreover, the

word piano covers actually two separate instruments, the grand piano and the upright piano.

They both are extremely complex from the physical point of view. This fact makes the

modeling task fascinating and challenging. The acoustics of the grand piano and the upright

piano have been an interesting research topic over decades, but despite the dozens of studies

the behavior of the instruments are anything but clear. An overview to the acoustics of the

grand piano is given in Chapter 2. This analysis can be applied mostly to the upright piano

as well.

An acoustical piano may not be suitable choice for all situations and for all pianists. For

example, a digital piano may be a better solution for those who live in apartment houses,

as the digital instrument does not cause structural noise, which travels from one apartment

to another. At the moment, most of the digital pianos use the sampling technique, in which

recorded piano tones are played back from the database. The advantage is that the sounds of

high-quality digital pianos are authentic and the computational requirements are not signif-

icant. On the other hand, the need for memory is heavy, since all tones must be saved in the

database with several different dynamic levels. Also, some features of the piano sound, such

as the coupling between the played tones, cannot be modeled correctly, since the tones are

simply played back separately from each other. These facts give a fruitful starting point to

the development of a parametric piano synthesis model where the basis lies on the physics

of the instrument. In addition, if the implementation of the model is done in a computation-

ally efficient way, physics-based sound synthesis can provide a good alternative for digital

pianos using the sampling technique. At the moment, the digital waveguide modeling has

turned out to be the most feasible physical sound synthesis technique for plucked and struck

string instrument modeling. It simulates the traveling wave solution of the wave equation

1

CHAPTER 1. INTRODUCTION 2

and provides a computationally efficient solution for the real-time implementation. The

digital waveguide modeling of the piano is studied in Chapter 3.

The piano has a very characteristic sound. The features of the piano sound must be taken

into account in the modeling procedure. Parametric modeling provides a good baseline

for a physical model, as the parameters can be extracted from recorded piano tones and

studied by means of different signal analysis tools. These parameters are used to control

the physical model, which imitates the sound production mechanism of the piano. The

main features of the piano tones are inharmonicity, which comes from the stiffness of the

piano strings, and complicated decay of the tones, which basically comes from the energy

leakage into the soundboard and other energy losses. A novel loss filter design technique

for modeling the decay of the tonesis presented in Chapter 4.

In addition to the string model, models for soundboard and sustain pedal are needed in a

high-quality parametric piano. Despite their importance, these are the least-studied parts of

the acoustic piano. Especially, only few models for the sustain pedal have been presented in

literature. An overview of the sustain pedal effect is given in Chapter 5, and a reverberator

algorithm is presented for modeling the effect.

Despite the fact that the piano has been studied by a number of researchers, the work

is by no means completed. It is curious that this instrument is very popular and known

widely around the world, but the physics of the instrument still remain partially unknown.

This affects also the modeling work. In order to make reliable physical sound synthesis,

the key assumption is that the physics of the instruments are known, at least at some level.

Moreover, the operation of human auditory system in the piano sound perception should

be known better in order to make the synthesis model more efficient. This is a part of the

future work, which is discussed in Chapter 6. This chapter also concludes the work.

Chapter 2

Overview of the Piano

The piano is probably the most popular instrument used in the western music. It has a very

characteristic sound and a wide dynamic range, and its playing range is more than seven

octaves. Usually the piano is used as a solo instrument but often it is used to accompany

other solo instruments or singing. Its popularity probably arises from its versatility and the

fact that it is capable of producing both powerful and sensitive sounds. Also, the control of

the sound production mechanism is relatively easy, as there are only three different control

parameters: The number of the key, the velocity of the press, and the pedals.

The roots of the modern piano go back into the beginning of the 18th century (Fletcher

and Rossing, 1991). In 1709, Bartolomeo Christofori of Florence modified the harpsichord

in order to make the instrument capable of variations in tone. He replaced the jacks with

hammers and called the new instrument the “gravicembalo col piano et forte”. During three

centuries, the instrument has evolved into two distinct instruments, the grand piano and

the upright piano. In Fig. 2.1 a Steinway & Sons grand piano is shown and in Fig. 2.2 a

Yamaha upright piano is presented. The pictures were taken in Espoo Music School during

the recording session in February 2005. For more information about the recordings, see

Appendix A.

The grand piano consists of five main parts, the keyboard, the action, the strings, the

soundboard, and the frame (Fletcher and Rossing, 1991). The keyboard consists of 88 keys

of which 52 are white and 36 are black. From the keystroke the information message is

transmitted to the action, which controls the hammer. The hammer hits the string and sets

it into vibration. The kinetic energy of the hammer is transformed into vibrational energy,

which is stored into the normal modes of the string. This energy is transmitted into the

soundboard via the bridge. The soundboard is a thin, wooden plate positioned under the

frame. The cast-iron frame, positioned to the upper part of the wooden case, keeps the

instrument together, and it is designed to withstand the high tension of the strings. The

3

CHAPTER 2. OVERVIEW OF THE PIANO 4

Figure 2.1: A grand piano.

Figure 2.2: An upright piano.

strings are attached to the tuning pins at the player end and to the hitch-pin rail at the other

end. The general structure of the instrument is presented in Fig. 2.3.

2.1 Keyboard

The player controls the output sound mainly with the keyboard. The choice of the key

determines the pitch of the tone, whereas the velocity of the finger press determines the

loudness. This interaction between the keyboard and the player is of great importance in


Key

Jack

Hammer Pin block

Damper

Soundboard

Ribs

String Bridge

Hitch pin

Joint hinge

Figure 2.3: The general structure of a grand piano.

the piano performance. Obviously, it is important that the feedback from the instrument to

the player is homogenous throughout the whole keyboard. The needed force for the key

depress should be the same for all keys. At the same time the “touch weight” should not

feel too light or too heavy to the pianist.

The “touch of the pianist” is a concept that divides the physicists’ and pianists’ opinions.

Physicists claim that the resulting sound does not depend on anything else than the velocity

of the finger press. On the other hand, many skilled pianists have the opinion that the

“touch” is an important factor in the art of piano music. By applying different kinds of

touches, the pianist can have major influence on the shadings of the music. It should be

pointed out that Askenfelt and Jansson (1990) have found out that at least the key motion

depends on the touch: The performance of the action is different depending on the playing

style, e.g., staccato and legato.

2.2 Action and the Hammer

The highly complicated machinery of the piano is called the action. This true specimen

of skill is responsible for transmitting the players intentions and accounts from the control

mechanism of the instrument to the strings. The most important part of the action is the

interaction between the hammer and the strings. Fig. 2.3 illustrates the principal idea of the

action and in the following, its operation is presented in general level. More comprehensive

studies can be found in (Askenfelt and Jansson, 1990) and in (Fletcher and Rossing, 1991).

Askenfelt and Jansson (1990) divided the piano action in four major parts: The key, the

lever body, the hammer, and the damper. When the key is depressed, the damper lying on

the string is lifted up. Right after this, the lever body is rotated upwards and the hammer

is accelerated when the jack pushes the roller positioned under the hammer. Just before the

hammer strikes the string, the jack reaches the escapement dolly and the top of the jack is

moved away from the hammer roller. The hammer continues the rest of its journey to the

string freely.


After the hammer strike, the rebounding hammer falls down and it is captured by the

check. This way the hammer is prevented from striking the string a second time, and the

action is now ready for repetition. If the key is held down after the hammer-string interac-

tion, the action will remain in this state. If the key is released right after the hammer stroke,

the action will return to its rest position.

The hammers have a strong effect on the tone quality. Their material, strength and size

contribute to the loudness as well as to the playability. The hammers are covered with two

layers of wool felt stretched over wooden moldings. The harder the felt material, the louder

and brighter the sound. In order to keep the loudness uniform throughout the scale, the

treble hammers must be harder than the bass hammers.

In modern instruments, the size and mass of the hammers decrease from bass to treble in

order to optimize the relation between aforementioned properties. If the hammer is heavy,

its contact time with the string increases. This, in turn, may increase losses in the case

that the roundtrip travel time for the initial pulse becomes shorter than the hammer-string

interaction time.

More information related to piano hammers and research dealing with hammers can be

found e.g. in (Conklin, 1990) and in (Conklin, 1996a).

2.3 Piano Strings

The strings are often called the heart of the piano. A concert grand piano has 243 steel-

made strings. The lowest strings are long and massive (length of even 2 m) while the

highest strings are thin and short. At the treble end, the length of the strings is only about

5 cm. At the bass end, the lowest strings overlap the middle strings providing them to get

nearer to the soundboard. The strings are terminated at the tuning pins at the player end and

at the hitch pin at the far end. The pins are made of steel and have a diameter from 3 to 4

mm. The speaking length of the string, though, is restricted to the bridge, see Fig. 2.3.

The first eight strings are single strings and the rest of the strings corresponding to the 80

highest keys are in groups of two or three strings, depending on the instrument. The strings

corresponding to the lowest 20 keys are wrapped with one or two layers of wire and the

rest 68 sets of three strings are unwrapped. Wrapping makes the bass string more flexible

compared to a solid string of the same diameter, which is advantageous since solid strings

with the same diameter and mass would result in highly inharmonic sound (Fletcher and

Rossing, 1991).


2.3.1 Inharmonicity

The inharmonicity in the piano strings is caused by stiffness. It makes the higher partials

travel faster on the piano string, which means that their frequencies are a little higher com-

pared to those of an ideal string. In the spectrum, the partial components are slightly shifted

making the series of overtones “stretched” upwards.

A mathematic expression for a stiff string can be obtained when the wave equation is

formulated as a combination of the equations for an ideal string and an ideal bar, see e.g.

(Smith III, 2004). Solving this equation for small stiffness values, a relation between the

modal frequencies and inharmonicity coefficient B can be solved. A more comprehensive

study on the mathematical derivation of this relation is given in section 3.2.1.

The inharmonicity coefficient of a tone can be measured, although it is far from being

an easy task. Galembo and Askenfelt (1999) presented a method which estimates the fun-

damental frequency and the inharmonicity coefficient at the same time. With this method

they reached convincing results, although the fundamental frequency estimation problem is

known to be challenging as well.

Usually, strong inharmonicity is considered to be an undesired feature of the piano sound.

On the other hand, it is not desired to get totally rid of it, since a slight inharmonicity adds

warmth to the sound (Fletcher et al., 1962). Interestingly, it has been shown that the human

ear is more sensitive to inharmonicity when the fundamental frequency of the tone is low

(Järveläinen et al., 2001). In this light, it is reasonable to use wrapped strings at the bass

range of the piano. Although the treble strings are highly dispersive, they contain only a

few partial components on the audio range. Therefore it is not necessary to eliminate the

effects caused by inharmonicity by using wound strings at the treble end of the piano.

The inharmonicity is taken into account in the tuning process (Fletcher and Rossing,

1991). The tuning is usually done in a way that the beating between the notes is minimized.

This will result in a stretched scale, where the fact that the upper partials of the lower tones

are slightly sharp with respect to the upper tones, is compensated. In other words, the lowest

strings are undertuned and the highest strings are overtuned.

2.3.2 Sound Decay

The sound generated by the piano strings is very complicated. The sound builds up rapidly

and decays slowly. Moreover, partials decay at different rates. Some of the partials may

sound even dozens of seconds whereas other partials decay in a few seconds. The spectrum

varies over time and differs from key to key; at the bass end over 50 partials can be extracted

while at the treble end the corresponding number is only about 3 or 4. As the strings

preserve most of the energy transmitted by the hammer and the action, the decay rate of the


tone depends on how rapidly the energy is leaking out from the strings.

The decay rate of the piano tone is twofold: The tone begins to decay rapidly, but after

some milliseconds the decay rate changes and becomes slower. As stated by Weinreich

(1977), the amount of aftersound, that is, the portion of the “slow” decay, depends on the

degree of mistuning. He also pointed out that the hammer irregularities may have an effect

on the two-stage decay as one string of the string group can be hit harder than another. There

is yet another explanation to the two stage decay: Change in the predominant vibration of

the strings. The vertical (perpendicular to the soundboard) vibration decays rapidly whereas

the horizontal (parallel to the soundboard) vibration decays slowly (Weinreich, 1977).

Other important phenomenon in the piano sound is the beating, which results from unison

groups of strings. When the hammer excites a tricord, that is, a set of three unison strings,

the strings begin to vibrate in the same phase. Due to small differences in frequency be-

tween the strings in the tricord, the tone starts to beat soon. The strings can be by no means

considered to be independent, since they are coupled to the bridge. This coupling allows

energy leakage between the strings resulting in a highly complicated system. A comprehen-

sive study about coupling of two strings and the bridge admittance is given in (Weinreich,

1977).

2.4 The Soundboard and the Frame

The soundboard is the main radiating structure of the piano. Usually it is made of spruce

strips which are glued together edge-to-edge. Widely used spruce species are Sitka spruce

or red spruce. Young’s modulus, as well as the strength, varies from tree to tree. These

values are much smaller in a direction at 90 degrees to the grain than in the grain direction.

This difference in cross-gain stiffness is compensated with ribs, which are glued onto the

bottom of the grand piano soundboard. In upright pianos, the ribs are glued onto the back

of the soundboard. Conklin (1996b) stated that an unribbed solid spruce soundboard would

cause splitting of the soundboard due to stresses caused by ambient factors, like changes in

humidity and temperature. He also stated that the output of the instrument would be less

than normal if there were no ribs on the soundboard.

The behavior of the soundboard can be examined with different kinds of methods. One

of the most popular is the Chladni method (Conklin, 1996b), which gives visual results, but

other powerful methods also exist. In addition, it is possible to investigate the vibrational

behavior of the soundboard by numerical analysis, like the finite element method. This

kind of approach was taken among others by Berthaut et al. (2003). The advantage of

computer simulation is that a prototype model of a soundboard need not be constructed, on

the understanding that the material parameters and the exact dimensions of the soundboard


are known.

The bridge couples the strings to the soundboard. Modern pianos have generally two

bridges, one for the bass strings and another for the middle and treble strings. Usually the

bridges are made of hardwood instead of softwood (like the spruce) which is unsuitable for

bridge material due to its low density. Too low a density allows extra losses at the string-

terminating points. Also, the strength of the softwood is inadequate to resist the mechanical

stresses around the metal bridge pins, where the strings are terminated. If the strings were

terminated straight to the soundboard, the resulting sound would be louder but unpleasant

and the decay times of the tones would be too short (Conklin, 1996b). Thus, the existence

and the quality of the bridge affects the piano sound remarkably.

Since the mechanical impedance of the soundboard is greater than the mechanical

impedance of the strings, the energy is leaking very slowly to the soundboard. An impor-

tant parameter which affects the leaking rate is the driving-point impedance, which varies

with frequency. The exact values can be determined by measurements. Logical places to

measure impedances are in several places on the bridges.

In the late 1800s, cast iron frames replaced the wooden string plates. This renewal en-

abled the increase of the acoustical output and extension of the keyboard. At the same time,

the quality of the piano sound improved. The cast iron plate provides proper strength to

support the high tension of the strings (which can exceed 1000 N!) as well as sufficient

stability (Fletcher and Rossing, 1991). Also the tunability improved as the tension of one

single string became almost independent of the tension of the other strings.

The cast iron frame is not a good tone radiator. Due to this fact, the string energy dissi-

pation within should be prevented. The plate should not vibrate during the playing and the

mechanical impedance of the plate should be substantially higher than that of the strings.

2.4.1 Pedals

The use of the pedals was not marked in musical notes before the 1790s. In the beginning,

it was considered gimmicky, but soon in the 19th century composers like Chopin and Liszt

took advantage of the modern technique and employed the usage of the pedals actively in

their work (Trustees of Dartmouth College, 2004).

With the help of the pedals, the player can add variation to the tones. Modern pianos

have either two or three pedals. The right pedal is always the sustain pedal, which lifts all

the dampers and sets all strings into vibration due to sympathetic resonance and the energy

transmission via the bridge. The sound rings as long as the sustain pedal is held down,

regardless of whether the key is released or not.

The left pedal is called the una corda pedal and its purpose is to make the output sound

softer. Una corda is Italian and means “one string”. In grand pianos, the whole keyboard


is shifted right causing the hammer to strike only one string instead of two, or two strings

instead of three. In addition to the decrease in loudness, the una corda pedal affects the

timbre as the coupling between the strings does not remain the same (Weinreich, 1977). On

the upright piano, the una corda pedal moves the action closer to the strings.

The purpose of the third pedal, which is positioned between the una corda and the sustain

pedal, can vary depending on the instrument. In some instruments, the center pedal can

work as a bass sustain. In this case, depressing the middle pedal sustains only the notes

in the bass section. In most grand pianos, the middle pedal is a “sostenuto” pedal which

sustains only the notes played before the pedal is depressed. This effect gives impression

of a third hand, because the player can employ both of his or her hands to the playing while

the sostenuto pedal keeps the chosen notes ringing. In some upright pianos, the center pedal

is a practice pedal. The tone is softened with a piece of felt which is lowered between the

hammers and the strings. (Piano World, 2005; Fletcher and Rossing, 1991)

2.5 Grand Piano vs. Upright Piano

A question that has long exercised the minds of researchers is whether there is audible

difference between the grand piano and the upright piano. It is clear that the structures

and mechanisms between these two instruments are different, but how these differences

correspond to the sounds of the instruments is still far from being completely obvious. In

the following, the structural differences are investigated and some aspects on the distinction

are given.

2.5.1 Structural Properties of the Upright Piano

Like grand pianos, there exist upright pianos of several sizes. The largest upright pianos

stand 150 cm in height whereas some studio uprights are only 100 cm of height. In large

upright pianos, the action is positioned some distance above the keyboard and in small

uprights the action is partly or even completely below the keyboard.

The major difference in the upright piano action compared to the grand piano is that

the hammers and the dampers move horizontally. The repetition lever, which enables fast

repetitions in grand pianos, is absent from the upright piano action. This means that the key

must be returned to its rest position before it can be depressed again. This is a clear feature

that most pianists recognize; it may affect also the prevailing opinion among musicians that

the grand piano is better than the upright piano. (Fletcher and Rossing, 1991)

The soundboard of an upright piano is rectangular. However, the vibrational area of

the soundboard is trapezoidal, because of the trimming rims that are positioned in the two

opposite corners of the soundboard. The two bridges lay diagonally on the soundboard,


inside the vibrational area.

2.5.2 Is There Noticeable Difference Between the Sounds of the Two Instru-ments?

A few studies exist about the distinction of the two instruments. Wogram and Mori (2003)

have listed the main properties of pianos with respect to their musical acoustical qualities.

They have measured these parameters with subjective listening tests and resulted in several

interesting observations. They used single tones, sequences of single tones, chords and

musical pieces in their listening tests, but found out that the sounds of upright and grand

pianos can only be distinguished reliably when musical pieces are played.

Probably the most interesting result is that according to their tests the two instruments

cannot be distinguished by the inharmonicity. This is surprising since usually the upright

piano is considered to have a more inharmonic sound than the grand piano as the strings

are, in general, shorter than in the grand piano. In the test, the subjects found the differences

in inharmonicity between the three strings of a choir in a grand piano greater than between

one upright piano string and the grand piano strings!

A more intuitive result concerns the sound radiation. Because of the different positions of

the soundboards of the two instruments, the listener perceives the sound fields in a different

way. It turns out that in the case of a grand piano there is a better balance between the bass

and treble sounds whereas in the case of an upright piano the bass seems to dominate the

sound. Another factor that affects the perceived sound balance is the eigenmode density of

the soundboard. In the case of the grand piano with higher eigenmode density, the sound is

better balanced.

As already mentioned, the piano action plays a key role in the distinction of the two

instruments for the player. Wogram and Mori (2003) found out that besides the ability

for faster repetition, the grand piano also has more efficient dampers. In the upright piano

action, the dampers are positioned beside the spot of attack, which results in insufficient

damping of some sound components. In grand pianos, the dampers are positioned in a

better way, and all modes will be damped properly.

2.6 Conclusion

In this chapter, an overview of the piano was given. This overview is by no means all-

inclusive, but rather a compact glance at the structure and acoustics of the piano. As it can

be noticed, the instrument is remarkably complex, but simple to control. This makes the

modeling task attractive and challenging at the same time. It is also clear that perceptual

issues can facilitate the modeling problem since there is no sense to model features that


exist but are not perceptually significant. This fact calls for reliable investigations on the

human perception and auditory system.

It is also worth noticing that the grand piano and the upright piano are two distinct instru-

ments. Although they sound similar to many people, their structures and sound production

mechanisms are different. Especially, skilled piano players and musicians distinguish the

instruments by both sound and the response of the instrument.

Chapter 3

Digital Waveguide Modeling of the

Piano

This chapter is divided into two major sections. In the beginning, a short overview of

synthesis methods is given. Since the digital waveguide (DWG) modeling technique is the

most important from the viewpoint of this thesis, it is presented in a more extensive way.

The second part concerns the digital waveguide modeling of the piano.

3.1 A Short Overview of Synthesis Methods

Smith (1991) proposed a taxonomy which divides the sound synthesis methods into four

different main classes: Abstract algorithms, processed recordings (and sampling), spectral

models and physical models. According to Tolonen et al. (1998), the last two methods

mentioned are the most promising when it comes to the future of sound synthesis. How-

ever, the two aforementioned methods give important basis to new, elaborate methods. For

a more comprehensive overview of synthesis methods, see references (Roads, 1995) and

(Tolonen et al., 1998). In the latter reference, also comparison between synthesis methods

is performed.

3.1.1 Abstract Algorithms

Synthesis methods that produce sounds based on some ad hoc mathematical formula are

considered to belong to the class of abstract algorithms. The most important synthesis meth-

ods belonging to this class are frequency modulation (FM) synthesis (Chowning, 1973),

waveshaping synthesis (Arfib, 1979; Le Brun, 1979) and synthesis based on the Karplus-

Strong (KS) algorithm (Karplus and Strong, 1983).

In general, with these methods it is almost impossible to produce natural sounds that

13

CHAPTER 3. DIGITAL WAVEGUIDE MODELING OF THE PIANO 14

resemble real instruments. On the other hand, the KS algorithm gives an important back-

ground to some physical sound synthesis methods (Smith, 1983; Jaffe and Smith, 1983).

3.1.2 Processed Recordings

Processing of recorded sounds (also called the sampling technique or wavetable synthesis),

is a sound synthesis technique in which sounds are played back from recordings (Roads,

1995). Its advantages are its rather easy implementation and the fact that this technique

offers a great variation of possibilities in terms of sound modifications. On the other hand,

the drawback is that the amount of memory required for implementation is huge. At the

moment, the best piano synthesizers are based on the sampling technique.

3.1.3 Spectral Models

Spectral models try to model and imitate the properties of sound waves. Well known meth-

ods of this class are, for example, additive synthesis, which is one of the oldest synthesis

methods, the phase vocoder (Flanagan and Golden, 1966) and the source-filter synthesis,

which is widely used especially in the area of speech synthesis (Moorer, 1985).

3.1.4 Physics-based Modeling Techniques

In the field of the sound synthesis of musical instruments, physical modeling is probably

the most widely used method at the moment. The strength of physical modeling is that it

provides intuitive approach to the control of the instrument models. The idea is to simulate

the sound production mechanism of the instrument instead of the sound itself. Physical

modeling techniques can be divided into six categories: Numerical solving of partial differ-

ential equations, source-filter modeling, vibrating mass-spring networks, modal synthesis,

wave digital filters and digital waveguide synthesis (Välimäki et al., 2006).

Numerical solving of the wave equation gives base to the finite difference method. This

approach was first taken by Hiller and Ruiz (1971a,b). This method is extensively used in

the field of sound synthesis of musical instruments. Its drawback compared to the digital

waveguide technique is that its computational complexity grows heavily with the size and

complexity of the model.

The source-filter synthesis mentioned in Section 3.1.3, can be included to the category of

physical models as well, because it models the sound source rather than the sound itself. The

vibrating mass-spring systems are based on the idea of modeling the mechanical elements

of a vibrating system, such as masses, dampers and springs. The idea was first presented by

Cadoz et al. (1983).


The idea of modal synthesis is that any system that produces sound can be divided into

vibrating substructures defined by modal data (Adrien, 1991). The energy flows between

the substructures via couplings and the system can also respond to external excitations. The

advantage of the method is that it can be applied to very complex structures.

Wave digital filters are initially developed for discrete-time simulation of analog electric

circuits. They are especially well suited for wave-based mass-spring modeling. This tech-

nique complements the DWG technique, as the nonlinearities in the instrument models can

be modeled with wave digital filters. The DWG technique is used for the wave propagation

part of the model. This technique was first presented by Fettweis (1986).

The DWG synthesis technique has turned out to be the most efficient and feasible when

it comes to the modeling of musical instrument sounds. The technique is especially well

suited e.g. to the simulation of a vibrating string or an acoustic tube. The theory is mainly

developed by Smith (1983, 1992, 1997). Also, in (Karjalainen et al., 1998) a very nice

introduction to the topic is given, in addition to some extensions. The idea of this method

originates from discretization of the traveling-wave solution of the wave equation. From this

point of view, the DWG modeling has the same starting point as the finite difference method.

However, it has certain advantages over the finite difference technique. For example, the

losses and the dispersion of the model can be lumped in one point, which means, in turn

that the whole system can be described with delay lines and linear filters.

In the following, the theory of DWG method is presented as a part of the piano string

model presentation as it describes the mathematical background for the method sufficiently.

3.2 Digital Waveguide Modeling of the Piano

Several DWG piano models can be found in literature. The first one was presented by

Garnett (1987). Later, in 1995, Smith and Van Duyne presented a piano model (Smith and

Van Duyne, 1995; Van Duyne and Smith, 1995) based on commuted synthesis. A more

recent study on DWG piano modeling was given by Bank (2000b). A good overview of

prior research and extensions to the above-mentioned model was presented by Bank et al.

(2003). A very detailed piano model partially based on DWG modeling was presented by

Bensa (2003) and Bensa et al. (2003).

3.2.1 Modeling the String

The approach taken here follows the DWG modeling scheme starting from the time-domain

solution of the wave equation. The one-dimensional wave equation for an ideal string can

be written (Morse and Ingard, 1968)


��

� �� (3.1)

where � is the position along the string, � is the transversal displacement, � is time,�

stands

for string tension and is linear mass density of the string.

The wave equation can be interpreted as the Newton’s second law, � � �� , where � is

force, stands for mass and � for acceleration. In the case of an ideal string, the force is

given as a product of the string tension and the curvature of the string, and the right side of

the equation is the mass density times transverse acceleration. The traveling-wave solution

of the wave equation can be written as a superposition of two waves, one traveling to the

right and the other traveling to the left:

�� !�#" �%$ ��"&�!�'� (3.2)

where � � is the right-going wave and � $ is the left-going wave. It is assumed that both � �and � $ are twice differentiable functions. Their shape can be arbitrary.

The traveling-wave solution in Eq. 3.2 can be discretized by sampling the amplitudes of

the traveling waves at every�

seconds. This corresponds to the spatial sampling interval(, which is the distance the wave propagates in

�seconds, that is

(*) � � , where � is

the sound velocity. Since one spatial unit corresponds to one temporal unit and the whole

wave propagates through the whole string, the wave propagating either right or left can be

simulated with a single digital delay line. Respectively, both waves can be simulated with

two parallel delay lines, see Fig. 3.1. This digital waveguide model of an ideal string can

be written as

��,+-��/.0� � � � ��1�� "&� $ ��12" �'3 (3.3)

In Eq. 3.3, the following changes of variables are made compared to Eq. 3.2: �546� . � ( and �748�9+ � 1 � . Also,

�is suppressed since it multiplies all arguments.

y + ( nT ,0)

y - ( nT ,0)

z - M y + (( n - M ) T , MX )

y - (( n+ M ) T , MX )

y ( n T , m X ) y ( n T ,0 )

z - M

Figure 3.1: Two parallel delay lines for simulating two propagating waves.


The simulation made with a digital waveguide model is exact only at the sampling in-

stants. However, it is possible to model a continuous waveform by interpolation between

two adjacent points. Also, with the choice of the sampling period it is possible to contribute

to the precision of the simulation. In general, the theoretical lower limit to the sampling

frequency is twice the highest frequency occurring in the system. Equivalently, the waves

propagating along the string must be bandlimited. Otherwise aliasing would occur and the

reconstruction of the signal becomes impossible. This restriction is called the Nyquist con-

dition, see e.g. Oppenheim and Schafer (1975), p. 29. In CD-quality audio, the sampling

rate is 44.1 kHz.

It should be pointed out that so far only a one dimensional case is studied. In piano syn-

thesis, at least three dimensions should be taken into account. That is, the two transversal

vibration polarization, the horizontal and the vertical, and the third corresponds to longitudi-

nal waves. Actually, when the hammer strikes three strings instead of one, nine waveguides

per key should be included. However, this kind of approach is usually unnecessary, since

the result is sufficient with one waveguide per coupled string (Bensa, 2003).

Modeling the Losses

In an ideal string, the string terminations are assumed to be rigid. This means that the waves

reflect with the same amplitude; the force waves change their sign and the velocity waves

reflect with the same sign. This is not the case with real piano strings, since losses occur due

to energy transmission to the bridge and soundboard. Also sympathetic resonance between

the strings causes energy losses, as well as the drag by the surrounding air. These losses

have to be modeled in high-quality piano synthesis. This is traditionally done by inserting

a loss filter to the string model (Van Duyne and Smith, 1995; Bank and Välimäki, 2003).

Basically, the loss filter can reduce to one loop attenuation factor when all frequencies

are decaying at the same rate. However, this is not the case with real strings, since high

partials tend to decay more rapidly than lower ones. Moreover, the variations in the gain

specification between two adjacent partials can be significant. The desired gain values can

be determined from a real piano sound, since the relation between the decay time constant of

the � th partial and the corresponding filter gain value �� can be expressed as a closed-form

formula:

�� (3.4)

where �� is the fundamental frequency, and �� is the decay time constant.

Van Duyne and Smith (1995) proposed a lowpass filter to be used as a loss filter. This

lowpass filter is a part of a coupling filter, which also takes into account the coupling be-


tween the three strings of a tricord. This approach is simplified but efficient way of model-

ing the overall losses and coupling between the strings. The problem in the model is that it

does not take the variation in the gain specification into account.

Bank (2000b) presented a transformation method for loss filter design. This technique

comes from the idea of using a certain transformed specification, which minimizes the error

of the decay times in a mean-square sense. The actual filter design process can be done by

any least squares filter design algorithm. Later, Bank and Välimäki (2003) introduced a

weighting function based on the first-order Taylor series approximation of the decay time

errors which can be used in the design of high order loss filters. Generally, both of these

aforementioned methods result in filters of relatively high orders.

Välimäki et al. (2004) proposed a combination of a one-pole filter and an FIR comb filter

called the ripple filter in order to achieve a reduced-complexity loss filter for harpsichord

synthesis. This technique allows exact matching of the decay rate of one partial and thus

some variation in overall response. However, this technique seems to be insufficient for

piano synthesis where the decay process is more complicated.

Rauhala et al. (2005) proposed an extension to the design method presented in (Välimäki

et al., 2004) in a sense that more than one feedforward path is added in cascade with the

one-pole filter. The design method is custom-made in addition to automatic smoothing of

the gain specification. It is capable of high accuracy with a low computational cost.

Modeling the Dispersion

Another physical phenomena occurring in piano strings is dispersion. Due to the high

stiffness of the strings, the resulting sound is inharmonic as the partials are slightly higher

than those of the harmonic case. This is a feature that can not be ignored in high-quality

synthesis. Unfortunately, for accurate simulation, a high-order filter is needed.

The inharmonicity of the strings is usually modeled with an allpass filter, which does not

affect the magnitude response of the system (Smith, 1983; Garnett, 1987). As the target is

to make the higher frequencies propagate faster along the string, a filter with a proper phase

response is needed. The frequency of the � th partial of a tone can be computed as (Fletcher

et al., 1962)

� � � � � �� "�� (3.5)

where � � is the nominal fundamental frequency and � is the inharmonicity coefficient. The

phase delay specification for the allpass filter can be written as

��

�� '� (3.6)


where is the delayline length of the string and� � � � � � is the delay specification of the loss

filter. From Eq. 3.6, the phase specification for the allpass filter can be derived.

Van Duyne and Smith (1994, 1995) proposed a bank of first-order allpass filters for the

dispersion simulation. However, this approach does not allow completely accurate simula-

tion of the dispersion phenomenon. A more accurate solution was introduced by Rocchesso

and Scalcon (1996). The method computes the filter coefficients from an overdetermined

system of equations by using the least-squared equation error criteria. In addition, the

method takes the fine-tuning of the string into account in the design process. The disadvan-

tage is the increased computational load.

Bank (2000b) introduced a multirate approach to decrease the computational load of the

dispersion simulation significantly. He took advantage of the fact that the lowest tones of

the piano can be simulated at half sampling rate. The filter simulating dispersion may thus

be more complex for accurate simulation of the phenomenon. On the other hand, only

interpolating filters in the outputs of the lowest tones are needed.

3.2.2 Modeling the Hammer

The interaction between the hammer and the string is highly nonlinear, which makes the

modeling procedure more complicated. The nonlinearity comes from the felt covering the

hammer, which exhibits a hysteretic behavior: The contact force during the compression is

different from that during the decompression.

The contact force between the hammer and string can be described by the equation

� �� (3.7)

where � is the hammer mass and � � is the hammer displacement. The relation can be

interpreted as the hammer being a lumped mass connected to a nonlinear spring. As can be

found from literature, see e.g. (Chaigne and Askenfelt, 1994a), the hammer-string interac-

tion can be described by the equation

� ��

�� (3.8)

where��!�� presents the compression of the hammer felt and � � is the

string position. In Eq. 3.8,�

stands for hammer stiffness coefficient and � is the stiffness

exponent. Both of these parameters are usually determined from experimental data. The

condition�� applies when the hammer is in contact with the string and the latter

condition�� is valid when the hammer is not in touch with the string.

Eq. 3.8 does not describe the phenomenon fully satisfactorily, since in real piano the


hammers indicate hysteretic behavior. This problem was studied by Stulov (1995). He

solved the problem by replacing the first part of Eq. 3.8 by

� ��

�� (3.9)

where �� $�� is a relaxation function, stands for hysteresis constant and � is

“nondimensional time”. As the relaxation function depends on time, it can be interpreted

that it represents the “memory” of the material.

When it comes to modeling, the hammer model can be discretized and coupled to the

string. There still exists a problem, though. There is a mutual dependence between Eq. 3.7

and Eq. 3.9, namely the hammer position should be known before computing the force and

the force should be known before computing the hammer position.

The implicit relation between Eq. 3.7 and Eq. 3.9 can be made explicit by inserting a

fictitious delay element in the model. This kind of approach is widely used in literature, see

e.g. (Chaigne and Askenfelt, 1994a,b). However, this kind of approximation can make the

model unstable.

To avoid the problem described above in addition to the nonlinearity problem, Smith and

Van Duyne (1995) came up with an idea that the hammer-string interaction consist of a

few discrete events during the hammer strike. These hammer strikes can be approximated

with one or more impulses that are filtered with a lowpass filters. Taking advantage of the

superposition of single lowpass filtered impulses the resulted signal approximates the force

impulse in a very efficient way. The parameters of these filters depend on the collision

velocity. That is, changing the collision velocity means that the filter parameters must be

changed. In addition, with this method the hammer restrike cannot be simulated correctly.

On the other hand, the advantage is that the linearized hammer can be included to the

commuted piano model.

A more general method was presented by Borin et al. (2000). This method, called the “K

method” maps the interaction force � as a function of the linear combination of the past

values of the string and hammer positions as well as the interaction force. The advance

is that the instantaneous dependencies of the variables are dropped. For a more extensive

description of the method, see (Borin et al., 2000).

A more recent study is presented in Bank (2000b,a). The multi-rate hammer model

overcomes the stability problem by doubling the sample rate in order to achieve smaller

changes in the variables of interest. However, doubling the sample rate in the whole string

model would double the computational load as well. To overcome this problem, Bank

suggests that only the hammer would operate at the increased sampling rate.


3.2.3 Modeling the Soundboard

The soundboard is the main radiating part of the piano. Its task is to color and amplify the

sound as well as to create the sensation of presence. Despite its importance, it is the least

studied part of the piano.

The modeling is often done with feedback delay networks (FDN), which are known to be

efficient for room reverberation simulations due to their abilities to achieve high modal den-

sity, see e.g. (Jot and Chaigne, 1991). The problem with FDN systems is that the choice of

parameter values seems to have no correspondence in the real world. The delayline lengths

are somewhat arbitrary, only the lengths are preferred to be prime numbers in order to avoid

the unwanted coloration (Gardner, 1998). In addition, the modeling can be addressed as a

filter design problem. Due to high modal density, high filter orders must be used.

The first DWG piano presented by Garnett (1987) included a soundboard model with six

extra waveguides connected to the bridge at a single location. This simple, but efficient

model can be interpreted as a predecessor for the feedback delay network solution for the

soundboard modeling. Bank (2000b) proposed a soundboard model structure based on

FDN with shaping filters. These filters match the system to imitate the overall magnitude

response of a real piano as well as possible.

Välimäki et al. (2004) presented an efficient system for harpsichord soundboard mod-

eling based on the algorithm presented in (Väänänen et al., 1997). The model consists of

eight delay lines, loss filters and comb allpass filters. This structure results in an authentic

impulse response of a harpsichord soundboard.

Chapter 4

A Novel Loss Filter Design Technique

The purpose of the loss filter is to model the decay rate of the partials accurately. Usually,

the loss filter is a low order FIR or IIR filter (Smith, 1983; Bank and Välimäki, 2003).

For good accuracy, a high-order filter is needed because the variations of the magnitude

response from one partial to the next one cannot be matched with a low order filter. In loss

filter design, a major problem is that the desired magnitude response is usually specified on

a very narrow frequency band. For example, for the lowest keys of the piano, the partials

below 2 kHz affect the sound most, and it is difficult to accurately estimate the decay rate

of high-frequency partials. However, the sampling rate used in high quality piano synthesis

is usually 44.1 kHz. The traditional filter design methods cannot face the challenge that the

important frequency band is only 10 percent of the audio range.

Another major problem faced in the design process is that the gain values can vary sig-

nificantly between two adjacent data points. Especially, the loss filters designed for the

lowest key values would have to follow very dense and detailed gain specification. This is

not possible for filters of low order.

The loss filter design technique presented here1 is an extension to the multi-ripple loss

filter design technique presented by Rauhala et al. (2005). The design technique proposed

in (Rauhala et al., 2005) consists of two cascaded subfilters. A first-order all-pole filter takes

care of modeling the general trend of the piano tone’s decay rate and the�

th-order feed-

forward comb filter models the decay rate variations from one partial to the next one. The

principal difference in the loss filter structure presented here is that in addition to the parallel

feedforward structure it is possible to insert feedforward blocks in cascade. Also the design

of the loss filter differs from the technique presented in (Rauhala et al., 2005). The method

is related to the frequency sampling (Parks and Burrus, 1987) and the IFIR filter techniques

(Neuvo et al., 1984). The structure of the proposed loss filter is based on a cascade of sparse

1also published in Lehtonen et al. (2005)

22

CHAPTER 4. A NOVEL LOSS FILTER DESIGN TECHNIQUE 23

FIR filters, which are designed one after the other on subbands that are integer fractions of

the audio range. Finally, the loss filter is up-sampled for implementation.

The proposed filter structure has three subfilters: The equalizer, the anti-imaging filter

and the multi-ripple filter. The equalizer is responsible for the general trend of the piano

tone’s decay, the anti-imaging filter attenuates the image frequency responses caused by

upsampling, and the multi-ripple filter is designed to fit the data as well as possible.

The purpose of the equalizer and the anti-imaging filter is basically the same as the anti-

imaging filter in the IFIR technique. In the IFIR technique, it is important to eliminate the

image frequency responses that result from the compression of the frequency response. In

this case, it is not desired to completely get rid of the image frequency responses, since there

are partials on higher frequencies as well. Instead, we need to be sure that these partials do

not dominate the sound but are appropriately attenuated.

The block diagram of the�

th-order loss filter is shown in Fig. 4.1. The blocks �� and �� are the equalizer, the anti-imaging filter, and the multi-ripple filter, respectively.

The system of Fig. 4.1 can be implemented in a time-reversed form, since then the delay

blocks of the anti-imaging filter and the multi-ripple filter can share delay elements with the

long delay line of length in the waveguide model.

c

a b

r 1

r 2

r N

z - M 2 R 1

z - M 2 R 2

z - M 2 R N

z - 1 z - M 1

H 1 H 2 H

3

x ( n ) y ( n )

Figure 4.1: Block diagram of the�

th-order loss filter consisting of three subfilters: � � is

the equalizer, � � is the anti-imaging filter, and � � is the multi-ripple filter.


4.1 Equalizer Design

The equalizer is designed as a first-order FIR filter in a way that the magnitude response

matches two given data points, such as those at the fundamental frequency and at the

Nyquist frequency. The result should imitate the general trend of the piano tone’s decay

rate in the band 0 � 22.05 kHz. The phase is not linear, but since the changes in phase delay

response are not significant, about 0.05 percent of the delay line length with practical

parameter values, this does not cause any problems. The dashed line in Fig. 4.2 presents

the magnitude response of the equalizer.

0 5 10 15 20

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Frequency (kHz)

Filte

r ga

in

Figure 4.2: The magnitude responses of the subfilters presented in Fig. 4.1: The equalizer

(dashed line), the anti-imaging filter with � � �� (dash-dotted line), the multi-ripple filter

with � �� (solid line), and the resulting loss filter (thick line).

4.2 Anti-Imaging Filter Design

The anti-imaging filter has two important tasks. Firstly, it takes care of attenuating the

resulting image frequency responses and secondly, it can be used for easing the modeling

of the important frequency band.

The filter is designed on a reduced frequency band in the same manner as the equalizer. It

is important that the up-sampling factor � � is selected in a way that the anti-imaging filter

attenuates the frequency band below 5 kHz appropriately. It has been found out that factor


6 is sufficient for the lowest piano tones (key index�� ) whereas for the key indices

about 10 � 20 factor 2 works well. In the middle range, where the harmonics cover a larger

portion of the audio range, the anti-imaging filter is not essential. The resulting magnitude

response is presented in Fig. 4.2 with a dash-dotted line.

4.3 Multi-Ripple Filter Design

The multi-ripple filter design process presented here results in the same multi-ripple filter

structure as presented in (Rauhala et al., 2005). On the other hand, the design processes

are somewhat different since the design method presented here uses standard filter design

techniques.

The desired gain values can be determined from a real piano sound since the relation

between the decay time constant of the � th partial and the corresponding filter gain value

can be expressed with the closed form formula of Eq. 3.4.

The general trend of the obtained gain is decreasing at higher frequencies. This makes the

task harder for the filter design algorithms. However, this problem can be made easier by

multiplying the gain values with the inverse magnitude response values of the equalizer and

the anti-imaging filter. Hence, the gain values are around one. This effect is compensated

at the end with the equalizer and the anti-imaging filter.

The problem that the data covers only a small part of the audio range can be overcome by

critical down-sampling. When the frequency of the largest partial is � .�� , the up-sampling

factor for critical down-sampling is � �� floor

� � � � � � � � � � .�� and the new sampling

rate is �� .

The actual design method is based on frequency sampling (Parks and Burrus, 1987). First

the impulse response (and thus the corresponding FIR filter coefficients) is obtained from

the magnitude response by inverse discrete Fourier transform. After this, the�

largest val-

ues are chosen and the other coefficients are set to zero. When�

is chosen to be the exact

number of the partials, it is possible to design a filter which models the given frequency re-

sponse perfectly. When the impulse response is up-sampled, the result is a sparse FIR filter.

In the frequency domain, the up-sampling means that � � ��

image frequency responses

follow the original frequency response. The magnitude responses of all three subfilters in

the case of the exact fit at the lowest frequencies are presented in Fig. 4.2.

In the case of low filter orders, the fit in the data cannot be perfect, though, the�

largest

values of the impulse response fit the obtained magnitude response to the data best in the

least-square sense. In the loss filter design, it is usually desired that especially the highest

peaks are modeled accurately, since the gain values near the value one have the longest

decay times. This fact can be taken into account in the design by emphasizing the largest


gain values with a weighting function presented by Bank and Välimäki (2003).

4.4 Design Example: Perfect Match of 50 Partials

This example describes how to design a filter which matches the 50 lowest-order partials

for the key index 3 ( � � �� 3�� Hz).

In the loss filter design, the case of an exact fit to the data has been considered partic-

ularly problematic. Laroche and Meillier (1994) have presented a method for designing

a perfect match: When all harmonics are measured correctly, the loss filter can be imple-

mented so that every partial is modeled with its own resonator. The computational cost of

the implementation becomes large at low fundamental frequencies, when there are many

partials to be modeled. The approach taken here is somewhat different and it leads to a

computationally more efficient implementation.

The first phase of the design process is to determine the new, reduced sampling rate.

When the target is to model 50 lowest-order partials accurately for the key index 3, the

highest frequency in the data is 1543 Hz. This means that the up-sampling factor � � is

chosen to be 14 and the new sampling rate used during the design is� � � � � Hz � ��

Hz.

The second phase is to design the equalizer and the anti-imaging filter. The up-sampling

factor � � for the anti-imaging filter is chosen to be 6.

In the next phase, inverse DFT is applied and the obtained impulse response is made

minimum phase with Matlab’s ’rceps’ function (MAT, 2004). After this, 50 largest im-

pulse response values are selected to be the FIR filter coefficients. At the end, the impulse

response is up-sampled by factor 14 and a sparse FIR filter is obtained.

In Fig. 4.3 (a), it is seen that the resulted magnitude response (solid line) follows exactly

the gain specification (dots). Fig. 4.3 (b) presents the corresponding�� -times, i.e. the

time it takes for each harmonic to decay 60 dB. The dashed line presents a response of the

system without the anti-imaging filter. It can be seen that in the�� -domain there are large

values around 3 kHz, which do not follow the general trend of the gain specification (dots).

This observation states that the usage of the anti-imaging filter is necessary especially in

the case of low key indices. In the case the anti-imaging filter is used, the� � -time is only

about 3 seconds, and thus does not have a significant effect on the resulting sound.

4.5 Design Example: Low Order Sparse FIR Filter

When the selected filter order is low, some compromises in the fitting must be done. One

solution is to smooth the original data so that a few important points are preserved. Smooth-


0 200 400 600 800 1000 1200 1400 16000.9

0.92

0.94

0.96

0.98

1

Frequency (Hz)

Gai

n

30 100 300 1 000 3 000 10 0000

10

20

30

40

Frequency (Hz)

T60

(s)

(a)

(b)

(a)

(b)

(a)

(b)

Figure 4.3: (a) The gain specifications (dots), the magnitude response (solid line) and the

magnitude response without the anti-imaging filter (dashed line) of the designed system and

(b) the corresponding reverberation time as a function of log frequency.

ing the data simplifies the design task significantly as most of the minor details are ignored.

In the following, the data is smoothed in the same way as in (Rauhala et al., 2005): The

amplitude maxima, loop gain maxima, and local maxima and minima are chosen as the

points to be preserved, and these points are connected with a smooth polynomial func-

tion, which can be obtained by Matlabs’ ’polyfit’, ’polyder’, and ’interp1’ functions (MAT,

2004). The smoothing can be done also with Linear Predictive Coding (LPC), which is

known to model the spectral peaks efficiently. However, neither one of the smoothing meth-

ods presented here is perfect. As the performance of the loss filter design relies heavily on

the data smoothing, further investigations should be done in order to improve the expression

power of this loss filter design technique.

After applying the inverse DFT, the four largest impulse response values are selected.

These values can be optimized with the sparse weighted least squares technique presented

in (Tarczynski and Välimäki, 1996). The optimized filter coefficients are calculated with

(Parks and Burrus, 1987)


�� (4.1)

where�

is the DFT matrix (see e.g. (Mitra, 2002)),

is a diagonal weighting matrix

whose � th diagonal element is � � , and�

is the target frequency response vector. The

matrix�

is modified in a way that only the elements corresponding to the selected impulse

response values are nonzero (Tarczynski and Välimäki, 1996). The weighting function is

the same as presented in (Bank and Välimäki, 2003), and it can be written as

� � � � � �0�� $�� 3 (4.2)

An example design is presented in Fig. 4.4 with a thick line. The up-sampling factors,

the equalizer, and the anti-imaging filters of the previous design example are used, because

the gain specifications are again for the key index 3.

0 200 400 600 800 1000 1200 1400 1600

0.92

0.94

0.96

0.98

1

Frequency (Hz)

Gai

n

0 200 400 600 800 1000 1200 1400 16000

10

20

30

Frequency (Hz)

T60

(s)

(a)

(b)

Figure 4.4: (a) The gain specification (dots) and the results of three design examples: Low-

order loss filter presented here (thick), low-order loss filter presented in (Rauhala et al.,

2005) (dashed), and high-order conventional filter (dash-dotted). (b) The corresponding

reverberation times.


4.6 Results and Comparisons

In this section, the novel loss filter design technique is compared against the existing meth-

ods. The filters are the loss filter presented in (Rauhala et al., 2005) and a filter which is

designed using Matlab’s ’invfreqz’ function and the weighting function in Eq. 4.2. The

latter filter design process results in an IIR filter with a first order denominator and a nu-

merator of order 201. The data is the same as used in Sections 4.4 and 4.5. The results of

the comparison are presented in Fig. 4.4.

The characteristics of the proposed loss filter are shown in Fig. 4.4 with a thick line. It

can be seen that the filter magnitude response follows the gain specification quite accurately

at frequencies below 800 Hz. Also the highest peaks are modeled well. In the� � -domain,

which is more interesting from the perceptual point of view, the results are also quite suffi-

cient.

The loss filter presented in (Rauhala et al., 2005) did also well in the comparison. Its

magnitude response is presented in Fig. 4.4 with a dashed line. The high-order IIR filter

is also practically as good as the two other filters presented, but the order of the filter is 40

times larger.

Obviously, the two loss filters presented here and in (Rauhala et al., 2005) are better for

instrument synthesis purpose, since their computational cost is significantly smaller than

the cost of the high-order filter with the same performance.

The differences of the proposed filter and the filter presented in (Rauhala et al., 2005)

are minor when it comes to accuracy of the modeling. The computational costs are equally

large. The major difference and improvement lies in the design. Since the filter proposed in

this thesis uses standard filter design techniques, it is somewhat easier to design.

4.7 Conclusion

A new method of the loss filter design for piano synthesis purposes was presented. The

proposed filter structure is based on a cascade of sparse FIR filters, which are designed one

after the other on subbands and are then up-sampled for implementation. The strengths

of this method are its simplicity and good performance even with low filter orders. It is

also possible to design a filter, which models the given gain specification perfectly. In

comparison against a traditional, non-sparse filter, the proposed loss filter performs well

with a significantly smaller computational cost.

Chapter 5

Modeling of the Sustain Pedal

The importance of the sustain pedal can hardly be overestimated when it comes to profes-

sional piano performance. The decisions that the pianist makes about using the sustain pedal

affects the whole style and nature of the music. At the same time, the effect is extremely

complex, both from the musical and the physical point of view.

The art of using the sustain pedal is far from being an easy subject, since the proper use of

the device is highly dependent on the era during which the music was composed. In the 18th

century the use of the sustain pedal was considered to be mainly a special effect, whereas

the piano music composed in the 19th and 20th centuries tends to be heavily pedaled. This

fact, among others, needs to be taken into account when it comes to high quality piano

music.

Despite the importance of the sustain pedal effect, the subject is much less studied than

most other areas of the piano performance and physics. Only few studies can be found

in literature (De Poli et al., 1998; Ambrosini et al., 1995), but they show interesting and

beyond dispute important results. In Section 5.1.2, a brief overview of the research done in

the field is given.

The use of the pedal device has primarily two purposes. First, it serves as “extra fingers”

in situations where legato playing is not possible with any fingering. Second, since all

strings are free to vibrate, a great enrichment of the tone is obtained.

5.1 Overview of Prior Work

In the first article that concerns the digital waveguide synthesis of the piano (Garnett, 1987),

the author divides the model into five main sections: The hammer, the strings, the bridge, the

soundboard and the pedals. There has been a considerable amount of work concerning the

four aforementioned subjects, but the resonance pedal effect is much less studied. However,

30

CHAPTER 5. MODELING OF THE SUSTAIN PEDAL 31

many authors have mentioned the pedal device in their articles and considered a way to

model it (see e.g. Garnett (1987); Bank et al. (2003)) but very few implementations have

been presented. Accordingly, there is a need for research in this particular field.

5.1.1 The Sustain Pedal Effect

The sustain pedal effect is twofold. Firstly, the strings corresponding to the depressed key

continue to vibrate after the key is released. However, this does not affect the reverberation

time significantly, as is shown in Section 5.2.1. Secondly, the other strings are set into

vibration by sympathetic resonance. Also, the impulse that the hammer gives to the strings

excites the string register (that is, the whole set of strings). This effect is clearly audible in

the piano sound, especially when single tones are played.

Due to the sympathetic resonance of the strings, beating in the tone is increased. To

analyze this phenomenon exactly is extremely difficult since the number of strings in a

concert grand piano is nearly 250. The total amount of harmonic components in the whole

system is huge, and every one of them can (at least, in theory) start to beat with one of the

harmonics present in the played tone. The beating is thus by no means regular as in those

cases the sustain pedal is not employed. This feature is shown in Figs. 5.5, 5.6 and 5.7.

In order to make the algorithm perform efficiently in the cases where many tones are

played at the same time, these phenomena must be somehow simplified. In addition, the

target is to find a simple solution which does not require any additional information other

than the key index of the tone.

5.1.2 Studies Concerning the Modeling of the Sustain Pedal

Two studies about the sustain pedal modeling can be found in literature. One of them is a

patent, which describes a pedal resonance effect simulation device for digital pianos (De

Poli et al., 1998). The other study is given by Ambrosini et al. (1995). Both of these

published solutions employ a certain number of simplified string models as a main part of

the sustain pedal algorithm.

De Poli et al. (1998) presented a model with 28 ideal string models, of which 18 are fixed

length and 10 are variable length. The output from these two junctions are lowpass filtered

with a cutoff frequency of 1 kHz. The output from the whole system is multiplied with a

coefficient, which determines the depth of the resonance pedal effect. Finally, the pedaled

sound is added to the direct sound. With this device, an authentic resonance effect can be

achieved.

Ambrosini et al. (1995) presented a little bit simpler model for sustain pedal simulations

applied to recorded piano sounds. The effect is obtained with a bridge-string model which


employs two parameters: One for controlling the decay time, that is, the loop gain, and the

other for the characteristic admittance, which controls the energy leakage from the strings

to the bridge and the soundboard. In this model, the 48 strings used are divided into two

groups: The first set of 24 strings consists of fixed length strings simulating the “ensemble”

of the strings whereas the other set consists of strings whose length is varied depending on

the depressed key.

5.2 Reverberator Algorithm for Sustain Pedal Modeling

In this section, a simple but efficient algorithm for sustain pedal simulation is presented. It

is based on 12 long delay lines which simulate the first 12 strings of the piano. In order to

develop the algorithm further, more information about the phenomenon must be achieved.

This analysis can be done by examining recorded tones and string register responses.

5.2.1 Signal Analysis

By analyzing the string register responses (that is, the response of all strings when the

system is excited with the hammer) and tones with and without the sustain pedal depressed,

two major differences can be found. The first one is caused by the freely vibrating strings,

which are set into vibration by the hammer energy as they are not damped. The second

phenomenon is noticed as increased beating in the tone, as the energy of the vibrating string

set is leaking into other strings via the bridge. In addition, airborne sound has prominent

effect on the excitation of the string register. In this section, the phenomenon is illustrated

with the key indices 4 (note C1), 40 (note C4) and 76 (note C7), which represent examples

of bass, middle and treble tones, respectively.

String Register Response

The first phenomenon can be illustrated by two examples. The first one is the impulse

response of the string register system when the sustain pedal is depressed and the string

group corresponding to the depressed key is damped. In the other case, the situation is

similar except the sustain pedal is not depressed.

The string register responses are illustrated with 3D waterfall plots, see Figs. 5.1, 5.2 and

5.3. The plots are obtained by performing time-dependent frequency analysis with Matlabs’

’specgram’ function (MAT, 2004) to the first second of the recorded string response (the

effect of the hammer is excluded). The signals are scaled in order to avoid the possible

differences in signal levels, which are due to recordings. The scaling is performed by

dividing each signal by its power and setting the maximum value to 0.99. The frequency


analysis is done with 256 point FFT applying a Hann window of length 64 samples with 32

samples overlap.

In the case of key index 4, the differences are shown in Fig. 5.1. In this figure, the

magnitude of the first second of the string register response is presented as a function of

frequency. As can be noted from Fig. 5.1 (a), after one second the overall response has

decayed about 25 dB, whereas in the other case (b) the response has decayed about 40 dB

within the same time. The situation is similar with the key indices 40 and 76, see Figs.

5.2 and 5.3. With these observations, it can be concluded that the reverberation time of the

string register system is longer when the sustain pedal is used.

(a) (b)

0

0.5

1

0

5

10

−40

−20

0

20

40

Time/sFrequency/kHz

Mag

nitu

de/d

B

0

0.5

1

0

5

10

−40

−20

0

20

40

Time/sFrequency/kHz

Mag

nitu

de/d

B

Figure 5.1: 3D plot of the string register response with (a) and without (b) the sustain pedal.

The key index is 4.

(a) (b)

0

0.5

1

0

5

10

−40

−20

0

20

40

Time/sFrequency/kHz

Mag

nitu

de/d

B

0

0.5

1

0

5

10

−40

−20

0

20

40

Time/sFrequency/kHz

Mag

nitu

de/d

B

Figure 5.2: 3D plot of the string register response with (a) and without (b) the sustain pedal.

The key index is 40.


(a) (b)

0

0.5

1

0

5

10

−40

−20

0

20

40

Time/sFrequency/kHz

Mag

nitu

de/d

B

0

0.5

1

0

5

10

−40

−20

0

20

40

Time/sFrequency/kHz

Mag

nitu

de/d

BFigure 5.3: 3D plot of the string register response with (a) and without (b) the sustain pedal.

The key index is 76.

Another way to illustrate the first phenomenon is to plot the energy of the string register

response with and without the sustain pedal in octave bands. The energy is obtained from

the spectrum of the signal by summing up the output corresponding to a certain octave band.

The energy is measured from the first second of the signal, and the effect of the hammer hit

is minimized by excluding it in order to obtain the energy of the true string register response.

The signals are scaled in the same way in order to avoid possible differences between signal

levels in the recordings. The center frequencies of the octave bands are shown in Table 5.1.

The results are shown in Fig. 5.4 in the case of key indices 4 (a), 40 (b) and 76 (c). From

the figure it can be seen that the energy is greater in those cases the sustain pedal is pressed

down. In the cases (a) and (c), there is a clear difference between the signal energies but

in (b) the phenomenon is not that clear. Generally, this comes from the input signal level

adjustment during the recordings. In the case of key index 40, the levels vary significantly.

The scaling process facilitates the problem, but as the signal is amplified, the background

noise is also amplified. However, in general we can draw a conclusion that the energy of

the string register response is greater when the sustain pedal is used.

Beating

The increased beating can be studied by separating the harmonics from the recorded signals

by bandpass filtering. After a sufficient number of harmonics have been separated, the

envelopes of the harmonics are calculated. In Figs. 5.5, 5.6 and 5.7 the results are shown in

the case of three example tones, C1, C4 and C7, respectively. In the first case, harmonics

with indices 2-7 are plotted, since the fundamental frequency was lost in the background

noise. In the second case, that is, the note C4, first six harmonics are plotted. In the case


Table 5.1: Octave bands used in the string register response energy calculations.Octave band Lower cutoff Center Upper cutoff

index frequency/Hz frequency/Hz frequency/Hz

1 22 31.5 44

2 44 63 88

3 88 125 177

4 177 250 355

5 355 500 710

6 710 1000 1420

7 1420 2000 2840

8 2840 4000 5680

9 5680 8000 11360

of the note C7, only two first harmonics are plotted as the others were lost in background

noise.

As can be seen, the effect of the beating depends on the pitch of the depressed key. In

the bass range, the differences are minor whereas in the middle range the differences are

significant. Moreover, the increased beating seems to be dominating especially in the lowest

harmonics. In the treble range, the beating is not prominent in either case. From these

figures it can also be seen that the decay time of the tone does not depend on the usage

of the sustain pedal in most cases. With some harmonics this assumption does not hold,

as seen in Fig. 5.5 in the case of the fifth harmonic. This is probably because of energy

leakage from some other strings that have a harmonic precisely at the same frequency.

These harmonics amplify each other causing longer decay time.

To conclude, there are two significant effects resulting from the usage of the sustain

pedal. The first one is a "stir" resulting from the string register system as it is excited by the

hammer corresponding to the key that is played. Another important effect is the increased

beating, as the energy is leaking to the string register via the bridge and the airborne sound.

5.2.2 Algorithm for Modeling the Sustain Pedal Effect

The basic idea of the reverberator algorithm for sustain pedal modeling to be presented

lies on the imitation of the string register. Ambrosini et al. (1995) suggested the usage of a

prominent number of string models. By using reduced amount of strings, the computational

load can be facilitated. The authors suggest that the highest strings can be neglected as they

have only minor influence on the sound, whereas the lowest keys contain more harmonics

and are advantageous in that sense. According to the listening tests they conducted, they


(a) (b)

1 2 3 4 5 6 7 8 9

−70

−50

−30

−10

10

30

50

70

Octave Band Index

Ave

rage

Mag

nitu

de/d

B

1 2 3 4 5 6 7 8 9

−70

−50

−30

−10

10

30

50

70

Octave Band Index

(c)

1 2 3 4 5 6 7 8 9

−70

−50

−30

−10

10

30

50

70

Octave Band Index

Ave

rage

Mag

nitu

de/d

B

Figure 5.4: The energy of the string register response when the sustain pedal is depressed

(solid line) and when the sustain pedal is not depressed (dashed line). The key indices are

4 (a), 40 (b) and 76 (c).

ended up with a model of 48 strings. However, the number of strings must be still reduced

in order to get a computationally efficient algorithm.

In the model presented here, the amount of strings is determined to be 12, and their

lengths correspond to 12 longest strings of the piano. By doing this, a reverberating com-

ponent for every tone in the piano can be guaranteed, at least at some accuracy. In rever-

beration algorithm design for imitating room acoustics, usually the delay line lengths of

incommensurate numbers are preferred in order to avoid unwanted coloration in the sound

(Gardner, 1998). In the case of sustain pedal modeling, the objective is to design a “bad”

algorithm in that sense, since some coloration and excited modes are wanted.

By inserting a comb allpass filter into the feedback loops, a diffusing effect can be

brought about (Väänänen et al., 1997) in the system. In addition, Väänänen et al. (1997)

pointed out that when the sum of the outputs from the strings is fed back to their inputs,


0 5 10

−50

0Partial 2

Time/s

Am

plitu

de/d

B

0 5 10

−50

0Partial 3

Time/s

Am

plitu

de/d

B0 5 10

−50

0Partial 4

Time/s

Am

plitu

de/d

B

0 5 10

−50

0Partial 5

Time/sA

mpl

itude

/dB

0 5 10

−50

0Partial 6

Time/s

Am

plitu

de/d

B

0 5 10

−50

0Partial 7

Time/s

Am

plitu

de/d

B

Figure 5.5: The envelopes of the harmonics 2-7 of the piano tone C1 (key index 4) when

the sustain pedal is depressed (solid line) and without the sustain pedal (dashed line).

the modal density can be increased significantly. This is important, because in real situa-

tions the number of excited signal components is large, and thus a sufficient modal density

must be achieved with the simulation model as well. The block diagram of the system is

presented in Fig. 5.8. The choice of the parameters is discussed later.

In order to increase beating, a resonator can be added to the signal path. As noted in the

previous section, the increased beating is dominating in the lowest harmonics. Bearing this

in mind, it is reasonable to tune the resonator close to the fundamental frequency. If it is

desired to have even more beating, several resonators can be included to the signal path.

However, this increases the computational load as well. As already mentioned, the effects

and the behavior of the beating phenomenon is extremely complex and calls for further

studies.


0 2 4 6

−50

0Partial 1

Time/s

Am

plitu

de/d

B

0 2 4 6

−50

0Partial 2

Time/s

Am

plitu

de/d

B0 2 4 6

−50

0Partial 3

Time/s

Am

plitu

de/d

B

0 2 4 6

−50

0Partial 4

Time/sA

mpl

itude

/dB

0 2 4 6

−50

0Partial 5

Time/s

Am

plitu

de/d

B

0 2 4 6

−50

0Partial 6

Time/s

Am

plitu

de/d

B

Figure 5.6: The envelopes of the first six harmonics of the piano tone C4 (key index 40)

when the sustain pedal is depressed (solid line) and without the sustain pedal (dashed line).

0 2 4

−60

−40

−20

0Partial 1

Time/s

Am

plitu

de/d

B

0 2 4

−60

−40

−20

0Partial 2

Time/s

Am

plitu

de/d

B

Figure 5.7: The envelopes of the first two harmonics of the piano tone C7 (key index 76)

when the sustain pedal is depressed (solid line) and without the sustain pedal (dashed line).

Delay Line Lengths

The lengths of the delay lines are set to match the frequencies of the 12 lowest tones of the

piano according to the formula 5.1.

� round� �� (5.1)


R(z) Z -L1 A 1 (z)

Z -L12 A 12 (z)

g dl

g dl

g fb

g mix

x in y out

Figure 5.8: The block diagram of the sustain pedal algorithm.

where � � is the sampling frequency and � � is the fundamental frequency of the tone. The

rounding operation is performed, because without a fractional delay filter (see e.g. (Laakso

et al., 1996) for more information about fractional delay filters) only integer delay lines can

be implemented. It would be possible to insert a fractional delay filter into the model, but

this does not improve the resulting sound significantly. Actually, a rough approximation is

even better here, since in the real situation the resonating frequencies are likely to be only

near the harmonics of the played tone.

Allpass Filter Parameters

The transfer function of the � th allpass filter can be written as

� � �� "�� $�� " � � �� $�� (5.2)

where all the filter coefficients � � �� are set to 0.05 and the allpass delays � � are chosen to

be 8% of the delay lines � , as suggested in (Välimäki et al., 2004).

Other Parameters

The gain values � � � , �� and � .�� in the algorithm must be chosen carefully, since they

affect the stability of the algorithm and adjust the relations of the dry and reverberated

sound. The coefficient � � � determines the decay of the reverberated sound. In general, to

guarantee the stability, the condition � � � � ��

must be satisfied. In this case, the value � � �was chosen to be 0.999.


Väänänen (1997) suggests that the sum of the outputs of the parallel comb filters can be

multiplied with a coefficient which is a negative number whose absolute value is smaller

than 1. This solution is a simplified structure of the multiple feedback delay network, orig-

inally presented by Jot and Chaigne (1991). The feedback matrix in the structure presented

by Jot and Chaigne (1991) is now replaced by a single coefficient. In this case, the feedback

coefficient �� was chosen to be � 0.15.

The mixing coefficient � .�� is a very critical gain value, because it determines the re-

lation between the dry and reverberated sound. Too large a value can make the sound

unnatural while too a small value does not highlight the effect sufficiently. Practical studies

concerning this coefficient show that the value of the mixing coefficient depends on the key

index. In the bass range, the value is remarkably smaller than in the mid range, for exam-

ple. At this point of the research process, the mixing coefficient is tuned by hand, but in

the future some relation between the mixing coefficient and the key index could be derived.

This can be done for example by analyzing the recordings, especially the case where the

string group corresponding to the key to be pressed is damped while holding the sustain

pedal down. For further description of the recordings, see Appendix A. In practice, the

mixing coefficients for the example tones C1, C4 and C7 are set to be 0.4, 0.18 and 0.05,

respectively.

Resonator Parameters

The purpose of the resonator is to increase the beating in the sound. It is tuned near the

fundamental frequency � � of the tone. The coefficients for the second-order resonator� ��

can be determined by formulas given in (Regalia and Mitra, 1987):

� �� 3 � �� " � ��"�� "�� " � � � $ � " � 3 � �� " � ��"�� $ ��"��

�" � � � $ � " � � $ � (5.3)

where� �

�� "��

(5.4)

� � �� (5.5)

� �� (5.6)

� �� "�� 3 (5.7)�

is the notch bandwidth and � is the center frequency of the notch in radians. � determines

the gain at the center frequency. The choice of � depends on the desired beating frequency.

As in this case the aim is to increase beating, instead of modeling the beating exactly, the


accurate value of the beating frequency is not critical. Examinations on this subject have

shown that e.g. � 3 � Hz �� Hz works sufficiently well.

5.2.3 Results and Analysis of the Algorithm

The proposed algorithm has been analyzed with recorded sounds by doing informal listen-

ing tests. The performance of the algorithm depends on the key index. At the moment, it

works quite well for the bass and treble range. The most difficult part of the piano from

the sustain pedal modeling point of view seems to be the middle range. In this range, both

beating and the “stir” of the string register are clearly audible. At the same time, the quality

of these effects are crucial elements for the authenticity of the synthetic sustain pedal. Es-

pecially, natural beating seems to be hard to implement with only one resonator. Increasing

the amount of resonators will not solve the problem, as the computational load increases

unnecessarily when many keys are depressed simultaneously, which is very a common sit-

uation when playing piano. A solution to this problem must be found in the future.

In the case of bass tones, the algorithm works sufficiently well. As a matter of fact,

the differences between the recorded sounds with and without the sustain pedal are not

prominent. However, there is a slight difference in nuance of these two cases, and this

nuance is fairly well obtained with the proposed algorithm.

The quality of the synthetic pedal in the case of treble tones is sufficient. The stir from

the string register sounds fairly authentic. The beating is not prominent in the real pedaled

treble tones, and the result obtained with this sustain pedal algorithm is authentic also from

this point of view.

At the moment, the algorithm does not work satisfactorily for the synthetic tones. This

is probably due to the properties of the synthetic tones, as the results with recorded sounds

are fairly good. However, to make these results reliable, a more formal listening test must

be arranged in the future.

5.3 Conclusions

In this section an overview of an important feature of the piano sound, the sustain pedal, and

a reverberator algorithm for modeling the pedal was presented. This gives a starting point

to the future work, which contains some extensions to the model presented above as well

as the model for the partial sustain pedal. In addition, the effect of the sustain pedal in the

case of the lowest and highest tones must be analyzed more carefully in order to obtain high

quality synthetic sustain pedal. At this point, the algorithm is tested mainly with recorded

tones. In the future, more testing with synthetic tones must be performed. Also, a relation

between the mixing coefficient and the key index must be derived.

Chapter 6

Conclusions and Future Directions

In this thesis, an overview of the piano was given. First, the acoustics of the grand piano was

studied in order to give a good baseline for the physics-based modeling work. As the digital

waveguide modeling is the most feasible modeling technique for plucked and struck string

instruments, it was studied in more depth. The most important features of the piano sound

were considered, and an overview of some modeling solutions that have been presented

in literature was given. As there were two salient points in the piano sound synthesis that

needed to be developed further, the solutions to these problems were selected to be the main

target of this work. The first one was the modeling of the losses in the piano sound, and the

other was the development of the algorithm for the sustain pedal modeling.

In Chapter 4 a novel solution for loss filter design for modeling the complicated,

frequency-dependent decay of the piano tone with some practical design examples was

presented. With this design technique it is possible to obtain computationally efficient so-

lutions compared to the traditional design techniques, which usually result in high-order

filters. In addition, this loss filter technique is easy to use, as it employs standard filter

design techniques as a part of the solution.

In Chapter 5, the sustain pedal effect was studied. The analysis was done to the recorded

string register responses and pedaled piano sounds. It was noticed that the usage of the

sustain pedal affects the sound and this effect needs to be taken into account in the modeling

procedure. A reverberator algorithm for the sustain pedal modeling was presented and the

choice of the parameters was considered.

As a result, some important features of the piano sound were studied in more depth and

modeling procedures for these features were presented. However, the presented models

could be improved further, and more work related to these solutions needs to be done. This

is a part of the future work.

In the future, the work related to the physical modeling of the piano will continue. There

42

CHAPTER 6. CONCLUSIONS AND FUTURE DIRECTIONS 43

are many issues to be studied and learned about the instrument, and the synthesis model

needs to be updated as new features are discovered.

Two main directions can be specified. Firstly, as the features of the usage of the sustain

pedal are unknown, these must be studied more carefully, and maybe some new analysis

techniques for the recorded sounds must be found. Another good option is to collaborate

with professional pianists in order to study the control of the sustain pedal in more depth.

This may be even a fundamental starting point, as the usage is only rarely an on-off process.

How the output sound depends on the control technique is an essential question for the

synthesis model. There is no point to make a sustain pedal algorithm that cannot be used in

an authentic way, especially if the other parts of the model succeed in imitating the sound

production mechanism of the instrument.

The second direction is to analyze how human auditory system perceives the features in

the piano sound. Or, more precisely, what is important or even crucial to model and which

features can be excluded in order to reduce the computational complexity of the model.

Good examples are the criteria how the loss filter design should, or actually, can be modeled.

For example, if the harmonic decay time analysis is performed to recorded sounds from two

different instruments, it is likely that there are variations between the results obtained from

the different instruments. Still, both instruments sound like piano. If the data are different,

and still both cases sound good, there must be some clearance in the design limits. In this

case, it would be possible to perform statistical analysis in order to determine limits for

the decay times, and design filters based on these limits instead of measuring the harmonic

decay times for all tones and imitate the obtained data precisely. It may simplify the model

in addition that the design procedure may become more straightforward. These assumptions

can be applied to other areas of the sound synthesis of the piano as well. As the perceptual

issues become clearer it is likely that the physical model of the piano can be more efficient

and result in more authentic sound.

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Appendix A

Recordings

Basics

– The grand piano: Steinway & Sons Mod. C Sn. 559907

– The upright pianos: Yamaha Mod. U3AS Sn. 4558497 and Yamaha Hybrid piano,

Mod. MPX1Z Sn. 5895823

– Place: Espoo Music School, Espoo, Finland

– Dates: February 23, 2005 (Session 1), and August 19, 2005 (Session 2)

Recording equipment

� Microphones: AKG C 480 B

� Amplifiers: Behringer Eurorack MX602A

� PC

� Sound card: Digigram VXpocket v2

� Impulse hammer

Description of the recordings

The recording sessions were both performed in Espoo Music School in the class

Beethoven, which is their biggest rehearsal class. In the first recording session, the sounds

of Steinway & Sons grand piano and Yamaha upright piano were recorded whereas in the

second recording session, sound of Yamaha hybrid upright piano was also recorded. In all

cases, the sounds were recorded with two channels. An example of the microphone posi-

tioning is given in Fig. A.1. The figure is from the second recording session, where the

50

APPENDIX A. RECORDINGS 51

microphones are positioned 32.5 cm away from the strings of the grand piano, and the mi-

crophone corresponding to the left channel is in � � �angle and the microphone correspond-

ing to the right channel is in� � �

angle. In the case of the upright piano, the microphone

positionings are shown in Fig. A.2. Both microphones are positioned 35 cm away from

the soundboard. In the first recording session, the setups were similar, except in the case of

the upright piano, where the microphone corresponding to the left channel was positioned

above the instrument. However, this setup proved to give insufficient results, as the sound

was colored in a different way when listening the channels separately, and the microphone

positions were changed in the second session.

Figure A.1: Microphone positioning in the case of the grand piano in the second recording

session.

Performed Measurements: Recording session 1

1. Soundboard and string responses (Grand piano and upright piano)

� Air radiation response: The bridge was hit with the impulse hammer in the

bass, middle and treble range, 3-5 times/position. The recordings were carried

out with and without the sustain pedal. The air radiation response recording is

illustrated in Fig. A.3. The impulse hammer is hit to the bass end of the bridge.

2. Dynamics (Grand piano and uprigth piano)

� Selected notes (C1, C4, C7) with four different dynamics: piano (p), mezzoforte

(mf ), forte (f ), forte fortissimo (fff ). Two cases: Long and short playing styles.

3. Pedal (Grand piano and upright piano)


Figure A.2: Microphone positioning in the case of the upright piano in the second recording

session.

Figure A.3: Recording the soundboard response. The bass end of the bridge is hit with the

impulse hammer.

� Selected notes (C1, C4, C7) with sustain pedal depressed with four different

dynamics (p, mf, f, fff ).

� Selected notes (C1, C4, C7) with half sustain pedal depressed with four different

dynamics (p, mf, f, fff ).

4. Staccato

� Grand piano: Selected notes (C1, C4, C7) with staccato with four different

dynamics (p, mf, f, fff ). Three cases: Without, with and with half sustain pedal.


� Upright piano: Selected notes (C1, C4, C7) with staccato with four different

dynamics (p, mf, f, fff ). Two cases: Without and with half sustain pedal. In the

case ’full sustain pedal’, only one dynamic level (fff ) was played.

5. Chords, scales and musical pieces (Grand piano and upright piano)

� Chords: C major, E minor, G major and B major. Left hand plays an octave of

the fundamental note and right hand plays the full chord: Octave of the funda-

mental note, third, and quint. Two cases: With and without the sustain pedal.

� Chromatic scale through the whole keyboard.

� Musical piece: Jean Sibelius: Neilikka, Op. 85 Nr. 2.

6. Hammer (Grand piano and upright piano)

� Selected notes (C1, C4, C7) with the corresponding string group damped with

four different dymanics (p, mf, f, fff ).

7. Dispersion (Grand piano and upright piano)

� All notes played with one dynamic level (mf ).

8. Special cases (Grand piano) (Fogwall, 2004)

� Damper effect: The sustain pedal was depressed, after that a loud chord was

played, the chord was released, but the pedal was held down.

� While the sustain pedal was held down, the chord was silently pressed down.

After a while the pedal was released, but the chord was still held down.

� Loud chords were played and immediately after releasing the keys, the sustain

pedal was depressed.

� Sympathetic resonance: An up-down scale was played while holding the sustain

pedal down.

� String resonance: A chord was layed down without the hammer touching the

strings and some keys were pressed and released.

Performed Measurements: Recording session 2

1. Pedal, Part I

� (Grand piano and upright piano) Selected notes (C1, C4, C7) played with four

different dynamics (p, mf, f, fff ) with and without the sustain pedal.


� (Grand piano) Selected notes (A0, D1, G1, C2, F2, B2, ES2, AS2, CIS4, F4,

H4, E5, A5, D6, GIS6, C7, F7, H7) played with and without the sustain pedal.

2. Pedal, Part II: Special Cases

� (Grand piano and upright piano) Selected notes (C and G from every octave)

played while the sustain pedal was depressed and the string group correspond-

ing to the key was damped.

� (Grand piano) Selected notes (G2, E3, C4, G5, E6) played in three different

cases: Simulated sustain pedal with 12 lowest keys silently depressed, without

the sustain pedal and with the sustain pedal.

� (Grand piano) Selected notes (G2, E3, C4, E5, E6) played with long and short

playing style when three nearest neighbor from the both sides were silently

pressed down.

� (Grand piano) Selected notes (C3, C4, C5) played when all other C-notes were

silently depressed.

� (Grand piano and upright piano) Selected notes (C and G from every octave)

played while the sustain pedal was depressed and the string group correspond-

ing to the key was damped right after the key was depressed.

� (Grand piano) The string response was recorded while pressing the sustain pedal

down quietly and heavily.

3. Pedal, Part III: Chords and Scales

� (Grand piano) Selected chords (C-major, E-minor, G-major and B-minor) were

played with and without the sustain pedal in bass, middle and treble range of

the piano.

� (Grand piano) Chromatic scale through the whole keyboard.

4. Staccato

� (Grand piano) Selected notes (C1, C4, C7) with staccato with four different

dynamics (p, mf, f, fff ). The sustain pedal was not employed.

5. Hammer

� (Grand piano) Selected notes (C1, C4, C7) with the corresponding string group

damped with four different dymanics (p, mf, f, fff ).


� (Grand piano) Air radiation response: The bridge was hit with the impulse ham-

mer in the bass, middle and treble range, 3-5 times/position. The recordings

were carried out with and without the sustain pedal.

Analysis and Parametric Synthesis of the Piano Sound

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